Finite Group Databasebeta

← #6615076

Subgroups — #6615076

Order 512.

2105 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic, complemented
2C4×#547C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic, complemented
4#33146C212yesmaximal, complemented
5#29219C211yesmaximal, characteristic, complemented
6#28279C211yesmaximal, characteristic, complemented
7#27158C214yesmaximal, complemented
8#26656C214yesmaximal, complemented
9#24965C212yesmaximal, complemented
10#24700C212yesmaximal, complemented
11#24150C212yesmaximal, complemented
12#24116C212yesmaximal, complemented
13#22962C212yesmaximal, complemented
14#22795C211yesmaximal, characteristic, complemented
15#22766C212yesmaximal, complemented
16C4×#459C211yesmaximal, characteristic, complemented
17#21386C212yesmaximal, complemented
18C2×#547—21no—
19C2×#547C2211yescharacteristic
20C2×#532C2212yes—
21C2×C4×#41—21no—
22C2×#474C2212yes—
23C2×#474C2212yes—
24C2×#459C2211yescharacteristic
25C2×C4×D8C2212yes—
26C2×#431C2212yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2547—21no—
30#2537—21no—
31#2537C2212yes—
32#2432—22no—
33#2422—22no—
34#2422C2214yes—
35#2365C2211yescharacteristic
36#2362—21no—
37#2361C2212yes—
38#2359—22no—
39#2354C2211yescharacteristic
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2212yes—
42D4×C4⋊C4C2211yescharacteristic
43#2344C2212yes—
44#2339—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×C22⋊Q8—21no—
48C4×#123—22no—
49C4×#121C2212yes—
50C42×D4C2211yescharacteristic
51C42×D4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53#2099C2214yes—
54#2075C2214yes—
55#2067C2214yes—
56#2033C2214yes—
57#2033C2214yes—
58#1999C2214yes—
59#1990C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1988C2212yes—
63#1982C2212yes—
64#1982C2211yescharacteristic
65#1954C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1947C2211yescharacteristic
69#1947C2212yes—
70#1942C2214yes—
71#1934C2214yes—
72#1925C2212yes—
73#1923C2214yes—
74#1918C2214yes—
75#1918C2214yes—
76#1879C2212yes—
77#1879C2214yes—
78#1874C2214yes—
79#1874C2214yes—
80#1862C2214yes—
81#1852C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1823C2214yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1797C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2214yes—
93C4×#102C2214yes—
94C4×#102C2212yes—
95C22×#41—21no—
96C22×#41—22no—
97C22×D8C2312yes—
98#547—24no—
99#547—24no—
100#547—48no—
101#532—28no—
102#532—44no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#124—22no—
109C2×#123—22no—
110C2×#123—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—21no—
113C2×#121—22no—
114C2×#121—22no—
115C2×#121C2312yes—
116C2×#121C2314yes—
117C4×#41—48no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—22no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2314yes—
132C2×C4×D4C2311yescharacteristic
133C2×C4×D4C2311yescharacteristic
134C2×#117—21no—
135C2×#117—22no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C42—22no—
140C22×C42—21no—
141C22×C42C2312yes—
142#474—28no—
143#474—44no—
144#459—24no—
145C4×D8—28no—
146#431—44no—
147C2×#107C2314yes—
148C2×#107C2312yes—
149C2×#107C2314yes—
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2312yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C22⋊C8C2314yes—
163C2×C22⋊C8C2314yes—
164C2×C4×C8C2312yes—
165C2×C4×C8C2314yes—
166#397—24no—
167#394—24no—
168#394—24no—
169#391—21no—
170#391—21no—
171#391—22no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#389—21no—
177#389—21no—
178#389—22no—
179#389C2312yes—
180#389C2312yes—
181#389C2314yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—24no—
192#385—24no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#381—22no—
201#380—22no—
202#380—21no—
203#380—22no—
204#380—22no—
205#380C2312yes—
206#380C2314yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226C2×#95—22no—
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#336C2314yes—
238#336C2312yes—
239C2×#41—28no—
240C2×#41—48no—
241C2×#41—44no—
242C2×#41—44no—
243C2×#41—48no—
244C2×#41—84no—
245C22×Q8—21no—
246C22×Q8—22no—
247C22×D4—22no—
248C22×D4—24no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2×D411yescharacteristic
252C22×D4C2412yes—
253C22×D4C2411yescharacteristic
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—24no—
260C23×C4—21no—
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2412yes—
264C2×D8—24no—
265C2×D8—24no—
266C2×D8—42no—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C4⋊Q8—22no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—44no—
274C4⋊Q8—82no—
275C4⋊Q8#4114yescomplemented
276#127—22no—
277#127#4114yescomplemented
278#124—48no—
279#123—216no—
280#123—416no—
281C22⋊Q8—28no—
282C22⋊Q8—48no—
283#121—24no—
284#121—24no—
285#121—28no—
286#121—44no—
287#121—48no—
288#121—48no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×D4—28no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—24no—
297C4×D4—22no—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—42no—
301C4×D4—41no—
302C4×D4—44no—
303C4×D4—48no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4—48no—
307C4×D4—48no—
308C4×D4—44no—
309C4×D4D818yescomplemented
310C4×D4#4114yescomplemented
311#117—44no—
312#117—48no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—48no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—48no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—24no—
380#107—28no—
381#107—44no—
382C4⋊C8—22no—
383C4⋊C8—24no—
384C4⋊C8—22no—
385C4⋊C8—24no—
386C4⋊C8—42no—
387#102—24no—
388#102—24no—
389#102—24no—
390#102—28no—
391#102—24no—
392#102—44no—
393#102—44no—
394C22⋊C8—44no—
395C22⋊C8—44no—
396C4×C8—24no—
397C4×C8—44no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—21no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409#95C2×D412yes—
410#95C2×D412yes—
411#95C2×D412yes—
412C24—21no—
413C24—22no—
414C24C22×D412yes—
415#41—432no—
416#41—832no—
417C2×Q8—24no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×D4—28no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—48no—
443C2×D4—48no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—84no—
447C2×D4C2×D814yes—
448C2×D4C2×#4112yes—
449C2×D4C2×#4114yes—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—28no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—81no—
487C22×C4—84no—
488C22×C4—82no—
489C22×C4C2×D818yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C2511yescharacteristic
500D8—48no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—24no—
510C4⋊C4—28no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—48no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—81no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4C2×D814yes—
539C4⋊C4C2×#4112yes—
540C4⋊C4C2×#4114yes—
541C4⋊C42−1+412yes—
542C22⋊C4—28no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4—42no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—82no—
557C22⋊C4C2×D818yes—
558C42—24no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—21no—
565C42—21no—
566C42—41no—
567C42—42no—
568C42—44no—
569C42—42no—
570C42—44no—
571C42—42no—
572C42—44no—
573C42C2×D814yes—
574C42C2×#4112yes—
575C42C2×#4112yes—
576C42C2×#4112yes—
577C42H(5,𝔽2)12yes—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—24no—
586C23—41no—
587C23—42no—
588C23—42no—
589C23—41no—
590C23—41no—
591C23—42no—
592C23—41no—
593C23—42no—
594C23—42no—
595C23—41no—
596C23—41no—
597C23—42no—
598C23C22×D814yes—
599C23C23×D411yescharacteristic
600Q8—48no—
601Q8—88no—
602D4—28no—
603D4—22no—
604D4—24no—
605D4—24no—
606D4—22no—
607D4—48no—
608D4—416no—
609D4—41no—
610D4—42no—
611D4—42no—
612D4—41no—
613D4—48no—
614D4—816no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—216no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—416no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—48no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—816no—
663C2×C4—88no—
664C2×C4—88no—
665C2×C4—84no—
666C2×C4#52512yes—
667C2×C4#52512yes—
668C2×C4#54612yes—
669C2×C4#54612yes—
670C2×C4C22×D814yes—
671C2×C4C22×D812yes—
672C2×C4C22×D812yes—
673C2×C4C22×D812yes—
674C2×C4#57512yes—
675C2×C4#57712yes—
676C2×C4C22×#4111yescharacteristic
677C2×C4C22×#4111yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×H(5,𝔽2)11yescharacteristic
680C2×C4C2×2−1+411yescharacteristic
681C2×C4#58511yescharacteristic
682C8—42no—
683C8—42no—
684C8—44no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—44no—
694C22—42no—
695C22—42no—
696C22—44no—
697C22—44no—
698C22—82no—
699C22—81no—
700C22—82no—
701C22—82no—
702C22—81no—
703C22—81no—
704C22—82no—
705C22—81no—
706C22C2×#52511yescharacteristic
707C22C2×#54611yescharacteristic
708C22#348911yescharacteristic
709C22C23×D811yescharacteristic
710C22C2×#57511yescharacteristic
711C22C2×#57711yescharacteristic
712C22#358611yescharacteristic
713C4—22no—
714C4—22no—
715C4—24no—
716C4—22no—
717C4—42no—
718C4—42no—
719C4—44no—
720C4—42no—
721C4—41no—
722C4—42no—
723C4—82no—
724C4—82no—
725C4—81no—
726C4—84no—
727C4#310014yes—
728C4#329414yes—
729C4#347511yescharacteristic
730C4#348211yescharacteristic
731C4#351111yescharacteristic
732C4#351711yescharacteristic
733C2—44no—
734C2—82no—
735C2—84no—
736C2#3640411yescharacteristic
737C2C2×#310011yescharacteristic
738C2#6059911yescharacteristic
739C2C2×#329411yescharacteristic
740C2#6115911yescharacteristic
741C2#6209511yescharacteristic
742C2#6214511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36127C2semidirect
2. C4×#547C2semidirect
3. #33195C2semidirect
4. #33146C2semidirect
5. #29219C2semidirect
6. #28279C2semidirect
7. #27158C2semidirect
8. #26656C2semidirect
9. #24965C2semidirect
10. #24700C2semidirect
11. #24150C2semidirect
12. #24116C2semidirect
13. #22962C2semidirect
14. #22795C2semidirect
15. #22766C2semidirect
16. C4×#459C2semidirect
17. #21386C2semidirect
275. C4⋊Q8#41semidirect
277. #127#41semidirect
309. C4×D4D8semidirect
310. C4×D4#41semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#361272
C4×#5472
#331952
#331462
#292192
#282792
#271582
#266562
#249652
#247002
#241502
#241162
#229622
#227952
#227662
C4×#4592
#213862